L11n134

L11n133

L11n135

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n134 at Knotilus!

Link Presentations

[edit Notes on L11n134's Link Presentations]

Planar diagram presentation	X8192 X18,11,19,12 X3,10,4,11 X17,3,18,2 X12,5,13,6 X6718 X9,16,10,17 X20,14,21,13 X22,16,7,15 X4,20,5,19 X14,22,15,21
Gauss code	{1, 4, -3, -10, 5, -6}, {6, -1, -7, 3, 2, -5, 8, -11, 9, 7, -4, -2, 10, -8, 11, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{4}-u^{2}v^{3}+u^{2}v^{2}-u^{2}v-uv^{4}+2uv^{3}-uv^{2}+2uv-u-v^{3}+v^{2}-v+1}{uv^{2}}}$ (db)
Jones polynomial	$q^{7/2}-2q^{5/2}+3q^{3/2}-5{\sqrt {q}}+{\frac {4}{\sqrt {q}}}-{\frac {5}{q^{3/2}}}+{\frac {4}{q^{5/2}}}-{\frac {3}{q^{7/2}}}+{\frac {2}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-az^{7}+a^{3}z^{5}-6az^{5}+z^{5}a^{-1}+4a^{3}z^{3}-12az^{3}+4z^{3}a^{-1}+4a^{3}z-9az+4za^{-1}+a^{3}z^{-1}-az^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-2a^{4}z^{8}-3a^{2}z^{8}-z^{8}-a^{5}z^{7}+3a^{3}z^{7}+4az^{7}+10a^{4}z^{6}+12a^{2}z^{6}+2z^{6}+5a^{5}z^{5}-10az^{5}-5z^{5}a^{-1}-14a^{4}z^{4}-13a^{2}z^{4}-3z^{4}a^{-2}-2z^{4}-7a^{5}z^{3}-a^{3}z^{3}+19az^{3}+11z^{3}a^{-1}-2z^{3}a^{-3}+5a^{4}z^{2}+6a^{2}z^{2}+2z^{2}a^{-2}-z^{2}a^{-4}+4z^{2}+2a^{5}z-3a^{3}z-11az-6za^{-1}-a^{2}+a^{3}z^{-1}+az^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 8                   1 -1 6                 1   1 4               2 1   -1 2             3 1     2 0           2 3       1 -2         3 2         1 -4       2 3           1 -6     1 2             -1 -8   1 2               1 -10   1                 -1 -12 1                   1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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